158 History of the Theory of Numbers. [Chap, v 
5(0, N) by use of 
a.(a, N) =a,(0, N), &.(a, iV) =6,(0, N) -aa,(0, N). 
Thus a,6._i — a,_i6i = l, so that the area of OA.A,_i is 1/2 if the point 
Ai has the coordinates o,, 6,. All points representing terms of the same 
rank in all the series of the same base he at equally spaced intervals on a 
parallel to the x-axis, and the distance between adjacent points is the num- 
ber of units between this parallel and the a:-axis. 
A. Hurwitz-^^ apphed Farey series to the approximation of numbers 
by rational fractions and to the reduction of binary quadratic forms. 
J. Hermes-^^ designated as numbers of Farey the numbers ri = l, 72 = 2, 
X3 = 7^ = 3, To = 4, tq = t7 = 5, t8 = 4, . . . with the recursion formula 
T„ = r„_2^+r2^+i-n+i, 2''<n^2'+\ 
and connected with the representation of numbers to base 2. The ratios 
of the r's give the Farey fractions. 
K. Th. Vahlen-^^'' noted that the formation of the convergents to a 
fraction w by Farey's series coincides with the development of w into a con- 
tinued fraction whose numerators are ±1, and made an application to the 
composition of linear fractional substitutions. 
H. Made-'° apphed Hurwitz's method to numbers a+hi. 
E. Busche"^ apphed geometrically the series of irreducible fractions of 
denominators ^a and numerators ^b, and noted that the properties of 
Farey series {a = h) hold [Glaisher-®^]. 
W. Sierpinski^^^ used consecutive fractions of Farey series of order m 
to show that, if x is irrational. 
===« U=i^ ^ 2 2j 
Expositions of the theory of Farey series were given by E. Lucas,'" 
E. Cahen,-'^ Bachmann.^'^ 
An anonymous writer,^'^ starting with the irreducible fractions <1, 
arranged in order of magnitude, with the denominators ^ 10, inserted the 
fractions with denominator 11 by listing the pairs of fractions 0/1, 1/10; 
1/6, 1/5; 1/4, 2/7;. . ., the sum of whose denominators is 11, and noting 
that between the two of each pair lies a fraction with denominator 11 and 
numerator equal the sum of their numerators. 
*«8Math. Annalen, 44, 1894, 417-436; 39, 1891, 279; 45, 1894, 85; Math. Papers of the Chicago 
Congress, 1896, 125. Cf. F. Klein, Ausgewahlte Kapitel der Zahlentheorie, I, 1896, 
19^210. Cf. G. Humbert, Jour, de Math., (7), 2, 1916, 116-7. 
*«9Math. Annalen, 45, 1894, 371. Cf. L. von Schrutka, 71, 1912, 574, 583. 
^s'ajour. fiir Math., 115, 1895, 221-233. 
^'^Ueber Fareysche Doppelreihen, Diss. Giessen, Darmstadt, 1903. 
"'Math. Annalen, 60, 1905, 288. 
*"BuU. Inter. Acad. Sc. Cracovie, 1909, II, 725-7. 
»"Th6orie des nombres, 1891, 467-475, 508-9. 
*'^fil4ments de la theorie des nombres, 1900, 331-5. 
"'Niedere Zahlentheorie, 1, 1902, 121-150; 2, 1910, 55-96. 
i^'OZeitschrift Math. Naturw. Unterricht, 45, 1914, 559-562. 
